Factor the following expression: $7$ $x^2$ $-29$ $x+$ $4$
This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(7)}{(4)} &=& 28 \\ {a} + {b} &=& & & {-29} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $28$ and add them together. The factors that add up to ${-29}$ will be your ${a}$ and ${b}$ When ${a}$ is ${-1}$ and ${b}$ is ${-28}$ $ \begin{eqnarray} {ab} &=& ({-1})({-28}) &=& 28 \\ {a} + {b} &=& {-1} + {-28} &=& -29 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {7}x^2 {-1}x {-28}x +{4} $ Group the terms so that there is a common factor in each group: $ ({7}x^2 {-1}x) + ({-28}x +{4}) $ Factor out the common factors: $ x(7x - 1) - 4(7x - 1) $ Notice how $(7x - 1)$ has become a common factor. Factor this out to find the answer. $(7x - 1)(x - 4)$